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Review of Applications will begin on February 11,
2013. Award notifications will be made shortly thereafter.

All applicants must send transcripts (unofficial is fine) and two letters of recommendation from professors familiar with the applicant's abilities to the program director (see right side).

Eligibility

You must be a U.S. citizen or U.S. permanent resident to be considered.
You must be an undergraduate student during the program running dates.
Students from underrepresented groups, especially first generation college students and American Indians, are encouraged to apply.

Program Director:
Dr. Jennifer McLoud-Mann
The University of Texas at Tyler
3900 University Blvd
RBN 4008
Tyler, TX 75799
(903) 566-7327
jmcloud@uttyler.edu

Housing Costs

Students will be required to stay in the Patriot Village Apartments at a rate of $15 per day ($840 for the entire 8 weeks).

2013 Program Dates

June 3 through July 26
Students must arrive by June 2 for housing check-in.

Program Funding

It is anticipated that this program will be funded for the summers of 2011, 2012, and 2013. All awards made are contingent on funds from the NSF.

What do students do in the UT Tyler Mathematics REU?

Students will work on research in groups of three students that are each led by a faculty advisor. The research groups will be centered around three exciting areas of mathematical research.

Program Benefits:

Work closely with faculty in gaining valuable experience in the process of mathematical research. See research topics below.
A $4000 stipend to pay for housing, food, and entertainment.
Learn how to write and typeset a professional mathematics article using LaTeX.
Learn how to prepare and deliver an excellent talk on mathematics. Students will be funded to attend national conferences to present the results of their research experience.
Get information on the process of choosing and applying for graduate programs in mathematics.
Fun team-building social activities, such as
- Outdoor activities (Camp Tyler ropes course, canoeing at Lake Tyler)
- A trip to Austin to visit museums and other fun locations.
- A trip to Dallas to visit the aquarium
Students will have full access to The University of Texas at Tyler's facilities, including the Patriot Center (swimming pool, gym) and the library.

Summer 2013 Research Topics:

Knot Theory -- led by Dr. Jennifer McLoud-Mann

Knot theory is very accessible to undergraduate students, and Dr. McLoud has successfully mentored 5 undergraduate research projects in this area. Student authors have presented their work at regional/state meetings, and almost all have published or submitted manuscripts for publication. Several problems related to Dr. McLoud’s research interests remain open for investigation:

Cell Knots. Given a lattice L, a lattice knot K is a closed nonintersecting polygonal path
that consists of unit edges whose endpoints lie on the lattice. The minimum edge number of a knot K is the smallest number of unit sticks that can be used to form the knot K, denoted mK. Minimum edge numbers for trefoil knots have been found in many standard lattices. A natural Voronoi tessellation of space is associated with a given lattice, and the 3D Voronoi cells that tessellate can be used to construct a cell knot. The minimum number of cells to form a knot K is called the minimum cell number of K and is denoted MK. The connection between mK and MK for a given knot K can be studied. For the simple cubic lattice, MK = 2·mk. Preliminary results indicate that the same result holds for the simple hexagonal lattice. Exploring the connection between mK and MK in the face-centered cubic lattice and the body-centered cubic lattice, important to chemists, will be interesting and valuable.
Lattice Stick Number. For a given lattice knot K, instead of focusing on the minimum
number of edges, one can focus on the minimum number of sticks to create it (called lattice stick number). A stick is a maximum line segment in the knot consisting of one or many unit edges. Stick numbers in the simple cubic lattice and simple hexagonal lattice have been studied. Similar work in the face-centered and body-centered cubic lattices remains unexplored and seems feasible.
Composite Knots. Given two knots K and L, an operation can be performed to construct
the composite knot of K and L, denoted K#L. For a specific invariant such as the minimal
edge number, determining the relationship between mK, mL, and mK#L would be interesting. In the simple cubic lattice, the edge number of the composition of two knots can always be reduced by two. That is, mK#L ≤ mK + mL − 2. Students can explore the relationship between these minimal edge numbers in other lattices, such as the simple hexagonal lattice, face-centered cubic lattice, and the body-centered cubic lattice.

Tiling Theory -- led by Dr. Casey Mann

Tiling theory is the study of shapes (tiles) that can be used to cover the Euclidean plane (or possibly a different geometric space) in such a way that tiles to not overlap. Such a covering of the plane is called a tiling or a tessellation. Tiles and the tilings that tiles admit allow for a high degree of complexity, and tiling theory is used to categorize or otherwise understand these complexities.

For summer 2013, the tiling theory group will begin by studying some background material on tiling theory. The main source used will be Tiling and Patterns by B. Grunbaum and G. C. Shephard as well as a few relavant research articles. As it turns out, Tilings and Patterns is full of open problems that undergraduate students can understand and start working on immediately! For this reason, there will be lots of possible research topics for this group. To give you a sense of the kind of topics that might be pursued, here are a few specific topics.

Equitransitive and unilateral tilings of the plane by squares. Such tilings have been classified for tilings involving only 3 different sizes of squares. How about 4 different sizes?
Archimedean tilings and uniform tilings of compact surfaces. The Archimedean tilings and uniform tilings have been classified for the Euclidean plane. How about about on the sphere? Or the torus?
In summer 2012 this group solved the problem of classifying the uniform edge-c-colorings of the uniform planar tilings. How about a classification for the 2-uniform planar tilings?

Combinatorics on Words -- led by Dr. David Milan

Students working in this topic will be introduced to the subject of infinite square-free words and nonrepetitive vertex colorings of graphs. A word is said to be square-free if it has no subword that is repeated twice in a row. In 1906, Thue discovered an infinite square-free word on three generators. One related and highly active area is nonrepetitive vertex colorings of graphs. In this context, a vertex coloring is an assignment of a letter to each vertex. A simple path in a colored graph can then be thought of as a word. A vertex coloring is nonrepetitive, if all such words are square-free.

The minimum number of colors required for a nonrepetitive coloring of a graph G is called the Thue number π(G). For example, Thue’s result implies that π(P) = 3 for any path P. One stunning result gives the upper bound π(G) ≤ 16d² where d is the maximum degree of a vertex in G.

Students will consider some of the following questions:

Square-Free Words
Many results about nonrepetitive vertex colorings make use of square-free words with added properties. REU participants will consider questions such as:

For which values k do there exist infinite square-free words having no repeated letters over gaps of length k?
Is there a square-free word on 3 letters that avoids palindromes of length at least 5?

These questions could be solved in many ways. Students experienced in computer programming could write code that will help search for solutions. There are also combinatorial approaches similar to the method Thue used to find his results.

Graph Colorings
In addition to finding words with interesting combinatorial properties, students will look for applications to graph coloring problems. The proof that the Thue number of a tree is 4 led to a result called the leveling lemma. Answers to the above questions could be used in a similar way, suggesting the related questions:

Given square-free words on 3 letters that avoid long palindromes, what is a suitable generalization of the leveling lemma?
Given a new leveling lemma, what progress can be made on the question in about whether every graph admits a subdivision with Thue number 3?

Research Experience for Undergraduates in Mathematics